The second question that should be considered is, ``Is the perceptual color space continuous?'' A mapping between two metric spaces is continuous at a point if for every positive number there exists another positive number such that implies . Thus as , . The mapping f is called continuous on X if it is continuous at every point [Itô 1993, page 317,].
Suppose the space of photic stimuli is the metric space X and the perceptual color space is the metric space Y. The two abstract conditions for a continuous mapping of light stimuli onto perceptual color space can then be reformulated and expanded into the following points.
It is apparent that if we only use properties of the distributions of energy quanta as measured individually from each point in the visual field to define the metric in the space of photic stimuli that the mapping onto perceptual color space cannot be continuous. Consider the example of color constancy illustrated in Figure 8. Even a stimulus which produces the exact distribution of photon energy quanta as x can be mapped onto two different and perceptually distinguishable points in color space, that is .
Many researchers in visual perception have implicitly assumed that the mapping of the probability distributions of energy quanta of photons onto perceptual color space is continuous. This assumption does not hold if only the spectral characteristics of individual points in the visual field are used to define the space of photic stimuli. If a continuous mapping between the space of photic stimuli and perceptual color space is to be defined, it will be necessary to remove the one--to--many relationship implied by the current formulations of the space of photic stimuli.
It might be argued that the metric of just noticeable differences is not the proper metric for perceptual color space. However, if perceptual color space is a metric space, its metric must be some form of perceptual differences. The requirement remains for a re--representation of the space of photic stimuli in order remove the one--to--many relation to perceptual color space before a continuous mapping of visual stimuli onto perceptual representation can be defined.
Traditionally, researchers have attempted to sidestep the issue of the discontinuous nature of the mapping of photic stimuli onto perceptual color space by holding luminance constant. While this does remove one potential source of one--to--many relationships within the mapping (while ignoring others), it does not address the fundamental question of the nature of the mapping in a rich photic environment such as we are confronted with on a day to day basis. Illuminance contrast, both local and global, must be taken into account in any mapping between the photic world and perceptual space if that mapping is to claim to be continuous. Without a continuous mapping, a uniform color scale will not be possible.